### 56121 results for qubit oscillator frequency

Contributors: Korotkov, Alexander N.

Date: 2010-04-01

The **qubit** Rabi **oscillations** are known to be non-decaying (though with a fluctuating phase) if the **qubit** is continuously monitored in the weak-coupling regime. In this paper we propose an experiment to demonstrate these persistent Rabi **oscillations** via low-**frequency** noise correlation. The idea is to measure a **qubit** by two detectors, biased stroboscopically at the Rabi **frequency**. The low-**frequency** noise depends on the relative phase between the two combs of biasing pulses, with a strong increase of telegraph noise in both detectors for the in-phase or anti-phase combs. This happens because of self-synchronization between the persistent Rabi **oscillations** and measurement pulses. Almost perfect correlation of the noise in the two detectors for the in-phase regime and almost perfect anticorrelation for the anti-phase regime indicates a presence of synchronized persistent Rabi **oscillations**. The experiment can be realized with semiconductor or superconductor **qubits**....(Color online) Numerical (solid lines) and analytical (dashed lines) dependence of zero-**frequency** detector noise S a a 0 and cross-noise S a b 0 on the phase shift ϕ between the bias voltage combs for several values of the pulse width δ t a , b , and also for the harmonic biasing. Almost complete noise anticorrelation at ϕ = ± π indicate persistent Rabi **oscillations**....Analyzed system: a double-quantum-dot **qubit** measured by two QPC detectors, which are biased by combs of short voltage pulses with **frequency** Ω coinciding with the Rabi **frequency** Ω R . ... The **qubit** Rabi **oscillations** are known to be non-decaying (though with a fluctuating phase) if the **qubit** is continuously monitored in the weak-coupling regime. In this paper we propose an experiment to demonstrate these persistent Rabi **oscillations** via low-**frequency** noise correlation. The idea is to measure a **qubit** by two detectors, biased stroboscopically at the Rabi **frequency**. The low-**frequency** noise depends on the relative phase between the two combs of biasing pulses, with a strong increase of telegraph noise in both detectors for the in-phase or anti-phase combs. This happens because of self-synchronization between the persistent Rabi **oscillations** and measurement pulses. Almost perfect correlation of the noise in the two detectors for the in-phase regime and almost perfect anticorrelation for the anti-phase regime indicates a presence of synchronized persistent Rabi **oscillations**. The experiment can be realized with semiconductor or superconductor **qubits**.

Data types:

Contributors: Reuther, Georg M., Zueco, David, Hänggi, Peter, Kohler, Sigmund

Date: 2011-05-05

Sketch of the flux **qubit** (blue) coupled to a dc-SQUID. The interaction is characterised by the linear coupling g 1 , which depends linearly on the SQUID bias current I b , and the quadratic coupling g 2 . The SQUID with Josephson inductance L J is shunted by a capacitance C . The **frequency** shift of the resulting harmonic **oscillator** (green) can be probed by external resonant ac-excitation A cos Ω a c t via the transmission line (black), in which the quantum fluctuations ξ i n q m t are also present....We propose a generalisation of dispersive **qubit** readout which provides the time evolution of a flux **qubit** observable. Our proposal relies on the non-linear coupling of the **qubit** to a harmonic **oscillator** with high **frequency**, representing a dc-SQUID. Information about the **qubit** dynamics is obtained by recording the **oscillator** response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent **qubit** **oscillations**. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity....eq:in-out-total1. In an experiment, this can be achieved by lock-in techniques which we mimic in the following way : First, we focus on the associated spectrum ξ o u t ω depicted in figure fig:**qubit**-osc-phase-spectrum(b). It reflects the **qubit** dynamics in terms of two sidebands around the central peak related to the **oscillator** **frequency**, here chosen as Ω = 10 ω q b . The dissipative influence of the environment, modelled by a transmission line (see figure fig:setup), is reflected in a broadening of this peak. The corresponding **oscillator** bandwidth is given as 2 α Ω , where α denotes the dimensionless damping strength; see app:QME. Here, we recall that the **oscillator** is driven resonantly by the external driving signal A cos Ω a c t , that is, Ω = Ω a c . In the time domain, the sidebands correspond to the phase-shifted signal ξ o u t t = A cos Ω t - ϕ e x p t with slowly time-dependent phase ϕ e x p t . In order to obtain this phase ϕ e x p t , we select the spectral data from a **frequency** window of size 2 Δ Ω centred at the **oscillator** **frequency** Ω , which means that ξ o u t ω is multiplied with a Gaussian window function exp - ω - Ω 2 / Δ Ω 2 . We choose for the window size the resonator bandwidth, Δ Ω = α Ω , which turns out to suppress disturbing contributions from the low-**frequency** **qubit** dynamics. Finally, we centre the clipped spectrum at zero **frequency** and perform an inverse Fourier transform to the time domain. If the phase shift φ e x t was constant, one could use a much smaller measurement bandwidth. Then the outcome of the measurement procedure would correspond to homodyne detection of a quadrature defined by the phase shift and yield a value ∝ cos φ e x p ....Time-resolved measurement of coherent **qubit** **oscillations** at the degeneracy point ϵ = 0 . The full **qubit**-**oscillator** state was simulated with the quantum master equation eq:blochredfield with N = 10 **oscillator** states and the parameters Ω = Ω a c = 10 ω q b , g 1 = 0.1 ω q b , g 2 = 0.01 ω q b , A = 1.0 ω q b . The dimensionless **oscillator** dissipation strength is α = 0.12 . The resonator bandwidth is given by 2 α Ω = 2.4 ω q b . (a) Lock-in amplified phase ϕ e x p t (dashed green lines), compared to the estimated phase ϕ t (solid red line) of the outgoing signal ξ o u t t . Here, ϕ t ∝ σ x t [cf. equation eq:dr-osc-phase], which is corroborated by the inset showing that σ x t performs **oscillations** with (angular) **frequency** ω q b . (b) Power spectrum ξ o u t ω for the resonantly driven **oscillator** (blue solid line). The sidebands stemming from the **qubit** dynamics are visible at **frequencies** Ω ± ω q b . In order to extract the phase information, we apply a Gaussian window function with respect to the **frequency** window of half-width Δ Ω = 1.2 ω q b , which turns out to be the optimal value for the measurement bandwidth....We consider a superconducting flux **qubit** coupled to a SQUID as sketched in figure fig:setup. The SQUID is modelled as a harmonic **oscillator**, which gives rise to the Hamiltonian...In figure fig:**qubit**-osc-fid(a) we depict the fidelity defect δ F = 1 - F between ϕ e x p t and ϕ t as a function of the **oscillator** **frequency** Ω = Ω a c for different quadratic coupling coefficients g 2 . As expected, the overall fidelity is rather insufficient for small **oscillator** **frequency** Ω **oscillator** bandwidth is too small to resolve the **qubit** dynamics, i.e., if ω q b **oscillator** **frequencies**, we again observe an increase of the fidelity defect, which occurs the sooner the smaller g 2 . This latter effect, which is only visible for the smallest value of g 2 in figure fig:**qubit**-osc-fid(a), is directly explained by a reduced maximum angular visibility of the phase ϕ t ∝ g 2 / Ω . Thus, figure fig:**qubit**-osc-fid(a) provides a pertinent indication for the validity frame of our central relation ...If the **qubit** is only weakly coupled to the **oscillator**, and if the latter is driven only weakly, the ** qubit’s** time evolution is rather coherent (see section sec:sn on

**qubit**decoherence). For this scenario, figure fig:

**qubit**-osc-phase-spectrum(a) depicts the time-dependent phase ϕ t computed with the measurement relation...(a) Fidelity defect δ F = 1 - F for the phases ϕ t and ϕ e x p t and (b) time-averaged trace distance D ̄ between the density operators of a

**qubit**with finite coupling to the

**oscillator**and a reference

**qubit**without

**oscillator**. Both quantities are depicted for various coupling strengths g 2 in dependence of the

**oscillator**

**frequency**Ω . All other parameters are as in figure fig:

**qubit**-osc-phase-spectrum. ... We propose a generalisation of dispersive

**qubit**readout which provides the time evolution of a flux

**qubit**observable. Our proposal relies on the non-linear coupling of the

**qubit**to a harmonic

**oscillator**with high

**frequency**, representing a dc-SQUID. Information about the

**qubit**dynamics is obtained by recording the

**oscillator**response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent

**qubit**

**oscillations**. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity.

Data types:

Contributors: Xian-Ting Liang

Date: 2008-09-03

The evolutions of reduced density matrix elements ρ12 (below) and ρ11 (up) in SB and SIB models in low-**frequency** bath. The parameters are the same as in Fig. 1.
...The spectral density functions Johm(ω) (b) and Jeff(ω) (a) versus the **frequency** ω of the bath modes, where Δ=5×109Hz,λκ=1,ξ=0.01,Ω0=10Δ,T=0.01K,Γ=2.6×1011Hz.
...The evolutions of reduced density matrix elements of ρ12 (below) and ρ11 (up) in SIB model in medium-**frequency** bath in different values of Ω0, the other parameters are the same as in Fig. 1.
...Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO....The response functions of the Ohmic bath in (a) low and (c) medium **frequencies** and effective bath in (b) low and (d) medium **frequencies**. The parameters are the same as in Fig. 1. The cut-off **frequencies** for the two cases are taken according to Fig. 2.
...The sketch map on the low-, medium-, and high-**frequency** baths.
... Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO.

Data types:

Contributors: Rabenstein, K., Sverdlov, V. A., Averin, D. V.

Date: 2004-01-26

The profile of coherent quantum **oscillations** in an unbiased **qubit** dephased by the non-Gaussian noise with characteristic amplitude v 0 = 0.15 Δ and correlation time τ = 300 Δ -1 obtained by direct simulation of **qubit** dynamics with noise. Solid line is the exponential fit of the **oscillation** amplitude at large times. Dashed line is the initial 1 / t decay caused by effectively static distribution of v ....We have derived explicit non-perturbative expression for decoherence of quantum **oscillations** in a **qubit** by low-**frequency** noise. Decoherence strength is controlled by the noise spectral density at zero **frequency** while the noise correlation time $\tau$ determines the time $t$ of crossover from the $1/\sqrt{t}$ to the exponential suppression of coherence. We also performed Monte Carlo simulations of **qubit** dynamics with noise which agree with the analytical results and show that most of the conclusions are valid for both Gaussian and non-Gaussian noise....The rate γ of exponential **qubit** decoherence at long times t ≫ τ for ε = 0 and (a) Gaussian and (b) a model of the non-Gaussian noise with characteristic amplitude v 0 and correlation time τ . Solid lines give analytical results: Eq. ( e7) in (a) and Eq. ( e16) in (b). Symbols show γ extracted from Monte Carlo simulations of **qubit** dynamics. Note different scales for γ in parts (a) and (b). Inset in (b) shows schematic diagram of **qubit** basis states fluctuating under the influence of noise v t . ... We have derived explicit non-perturbative expression for decoherence of quantum **oscillations** in a **qubit** by low-**frequency** noise. Decoherence strength is controlled by the noise spectral density at zero **frequency** while the noise correlation time $\tau$ determines the time $t$ of crossover from the $1/\sqrt{t}$ to the exponential suppression of coherence. We also performed Monte Carlo simulations of **qubit** dynamics with noise which agree with the analytical results and show that most of the conclusions are valid for both Gaussian and non-Gaussian noise.

Data types:

Contributors: Strand, J. D., Ware, Matthew, Beaudoin, Félix, Ohki, T. A., Johnson, B. R., Blais, Alexandre, Plourde, B. L. T.

Date: 2013-01-03

Figure fig:FreqVsAmpl(a) shows linecuts of the experimental (black dots) and numerical (full red lines) chevrons. The linecuts are taken at the **frequency** ω F C corresponding to the maximum-visibility sideband **oscillations**, indicated by the full and dashed vertical lines in Fig. 3. The agreement between the experiments and simulations is excellent. In particular, the decay rate of the **oscillations** can be explained by the separately measured loss of the **qubit** and cavity and roughly corresponds to κ + γ 1 / 2 , where γ 1 is the bare transmon relaxation rate. This is expected for **oscillations** between states | e 0 and | g 1 . It also indicates that for these powers, the visibility loss can be completely attributed to damping. The lack of experimental points at pulse widths < 30 n s is a technical limit of the present configuration of our electronics that can be improved in future experiments.x x...(color online) (a) Schematic of energy levels in a combined **qubit**-resonator system, showing first-order red sideband transition. (b) Optical microscope image with inset showing expanded view of one of the **qubits**. The terminations of the flux-bias lines for both **qubits** are visible, and they are used for both dc bias and FC signals. (c) Schematic of **qubit**-cavity layout and signal paths....We used a sample consisting of two asymmetric transmon **qubits** capacitively coupled to the voltage antinodes of a coplanar waveguide resonator [Fig. fig:schem(b, c)]. The cavity had a bare fundamental resonance **frequency** ω r / 2 π = 8.102 G H z and decay rate κ / 2 π = 0.37 M H z . **Qubit**-state measurements were performed in the high-power limit . The **qubits**, labeled Q1 and Q2, were designed to be identical, with mutual inductances to their bias lines of 1 p H for Q2 and 2 p H for Q1. The **qubits** were excited by microwave pulses sent through the resonator, and the flux lines were used for dc flux biasing of the **qubits** as well as the high-speed flux modulation pulses for exciting sideband transitions. The dc flux lines included cryogenic filters before connecting to a bias-T for joining to the ac flux line, which had 20 / 6 / 10 d B of attenuation at the 4 K / 0.7 K / 0.03 K plates. The distribution of cold attenuators and the flux-bias mutual inductances were chosen as a compromise to allow for a sufficient flux amplitude for high-speed modulation of the **qubit** energy levels with negligible Joule heating of the refrigerator while avoiding excessive dissipation coupled to the **qubits** from the flux-bias lines....(color online) Spectroscopy vs. flux for Q2 showing g-e (solid blue points) and e-f (hollow red points) transition **frequencies**. Blue and red lines correspond to numerical fits. Heavy black line shows bare cavity resonance **frequency**. Vertical dashed line indicates flux bias point for sideband measurements described in subsequent figures along with ac flux drive amplitude, 2 Δ Φ = 70.9 m Φ 0 , corresponding to 2 Δ ω g e / 2 π = 572 MHz, used in Figs. 3(c), 4(c)....Figure fig:FreqVsAmpl(d) shows the sideband **oscillation** **frequency** Ω / 2 π extracted from the experimental linecuts (blackxdots) as a function of the flux-modulation amplitude Δ Φ . As expected from Eq. ( eq:H:t), whose prediction is given by the solid black line, the dependence of Ω with Δ Φ is linear at low amplitude and deviates at larger amplitudes. Beyond this simple model with only two transmon levels, quantitative agreement is found between the measured data and numerical simulations (full red line). For the numerical simulations, the link between the theoretical flux modulation amplitude Δ Φ and applied power is made by taking advantage of the linear dependence of Ω with Δ Φ at low power. Because of this, it is possible to convert the experimental flux amplitude from arbitrary units to m Φ 0 using only the lowest drive amplitude for calibration....We demonstrate rapid, first-order sideband transitions between a superconducting resonator and a **frequency**-modulated transmon **qubit**. The **qubit** contains a substantial asymmetry between its Josephson junctions leading to a linear portion of the energy band near the resonator **frequency**. The sideband transitions are driven with a magnetic flux signal of a few hundred MHz coupled to the **qubit**. This modulates the **qubit** splitting at a **frequency** near the detuning between the dressed **qubit** and resonator **frequencies**, leading to rates up to 85 MHz for exchanging quanta between the **qubit** and resonator....(color online) (a-c) Experimental data showing sideband **oscillations** as a function of pulse duration vs. flux-drive **frequency**. The amplitude of the flux pulse is reduced by (a) 10 d B , (b) 4 d B relative to (c). (d-f) Corresponding numerical simulations of sideband **oscillations** vs. drive **frequency**. Vertical white lines running through each plot indicate the **frequency** slices used in Fig. fig:FreqVsAmpl....(color online) (a),(b),(c) Sideband **oscillations** corresponding to the white slices in Fig. fig:chevron(a-c). Experimental points correspond to black dots; numerical simulations (not fits) indicated by red lines. (d) Sideband **oscillation** **frequency** vs. flux drive amplitude (lower horizontal axis) or corresponding **frequency** modulation amplitude (upper horizontal axis). The dashed line shows a linear fit to the low **frequency** data points, while the red solid line indicates the theoretical dependence from the numerical simulations. The full black line shows the analytical sideband **oscillation** **frequency** from Eq. ( eq:H:t). ... We demonstrate rapid, first-order sideband transitions between a superconducting resonator and a **frequency**-modulated transmon **qubit**. The **qubit** contains a substantial asymmetry between its Josephson junctions leading to a linear portion of the energy band near the resonator **frequency**. The sideband transitions are driven with a magnetic flux signal of a few hundred MHz coupled to the **qubit**. This modulates the **qubit** splitting at a **frequency** near the detuning between the dressed **qubit** and resonator **frequencies**, leading to rates up to 85 MHz for exchanging quanta between the **qubit** and resonator.

Data types:

Contributors: Jerger, Markus, Poletto, Stefano, Macha, Pascal, Hübner, Uwe, Il'ichev, Evgeni, Ustinov, Alexey V.

Date: 2012-05-29

An important desired ingredient of superconducting quantum circuits is a readout scheme whose complexity does not increase with the number of **qubits** involved in the measurement. Here, we present a readout scheme employing a single microwave line, which enables simultaneous readout of multiple **qubits**. Consequently, scaling up superconducting **qubit** circuits is no longer limited by the readout apparatus. Parallel readout of 6 flux **qubits** using a **frequency** division multiplexing technique is demonstrated, as well as simultaneous manipulation and time resolved measurement of 3 **qubits**. We discuss how this technique can be scaled up to read out hundreds of **qubits** on a chip....A probe signal composed of one microwave tone per **qubit** to be read out is sent through the common transmission line. The interaction between each **qubit** and resonator leads to a state-dependent dispersive shift, Δ ω r = g ~ 2 / ω q - ω r σ z , of the resonator **frequency**, where g ~ is the effective coupling between the resonator and **qubit**, ω q and ω r are the angular resonance **frequencies** of the **qubit** and resonator, and σ z is ± 1 depending on the state of the **qubit**. The composite signal probes all resonators at the same time, storing the information on the state of all **qubits** in the transmitted tones. Detection of the transmitted amplitude and phase of each of the tones provides a simultaneous non-destructive measurement of the states of all **qubits**. The probe signal is generated by mixing a reference microwave tone in the band of the resonators and a multi-tone DAC output using an IQ mixer, see Fig. fig:setup. By using the I and Q quadratures, we address the upper and lower sidebands of the mixing product individually to effectively double the bandwidth of the system. The mixer output is combined with the **qubit** manipulation signal through a directional coupler. A strongly attenuated line transmits the combined signal to the sample, which is attached to the mixing chamber stage of a dilution refrigerator. Two cryogenic circulators and a high-pass filter at 30 mK are used to prevent reflections and noise from traveling from the cryogenic amplifier back to the sample. A chain of amplifiers provides 80 dB gain to boost the transmitted probe signal to levels sufficient for the detection stage, which employs an identical IQ mixer to convert the signal back to baseband **frequencies**. The local **oscillator** inputs of both mixers are fed from the same reference microwave source, resulting in a homodyne detection with a fixed phase offset. An additional high-pass filter between the local **oscillator** ports of the two mixers prevents leakage of the baseband signal. After digitizing both quadratures, the amplitude and phase of all components of the probe signal are extracted via FFT. The maximum number of devices that can be probed with the described technique is defined by the **frequency** separation between resonators and the bandwidth of the acquisition board. The **frequencies** of the resonators on our chip are spaced at intervals of 150 MHz and the acquisition board has a bandwidth somewhat below 500 MHz, allowing for the simultaneous detection of up to six devices....In the next set of experiments, we tuned the uniform flux coil and two compact local coils placed above the sample to bias three **qubits** at their symmetry points. Limiting the number of **qubits** to three was necessary because of the lack of additional (on-chip) coils and not due to the readout technique itself. After setup of the readout pulse to probe circuits number #2, 3 and 5, we performed a spectroscopy of all three **qubits** simultaneously. A continuous microwave excitation signal of varying **frequencies** was applied to the sample and a pulsed three-tone probe signal was applied every 10 μ s . When the excitation **frequency** matches the gap between the ground and first excited states of a **qubit**, the instantaneous dispersive shift of the center **frequency** of the corresponding resonator switches between positive and negative, thus changing the mean amplitude and phase of the transmitted probe tone. Figure fig:multi_spec shows the spectra of three **qubits** measured in parallel....Finally, we performed simultaneous manipulation with time resolved measurements on three **qubits**. Here, we used individual microwave excitations for every **qubit**, which were added together via a power combiner. We note that the complete excitation chain could be replaced by a reference source and a mixer controlled by a single arbitrary waveform generator with sufficient bandwidth to drive all **qubits**, similar to FDM readout tone generation. Measurement data are reported in Fig. fig:rabi. All three **qubits** were simultaneously driven by individual excitation tones and the readout was performed in parallel using the FDM protocol. Every **qubit** can be Rabi-driven at a different power. Left panels of Fig. fig:rabi present Rabi **oscillations** at three different powers for all **qubits**. The measured linear power dependences of Rabi **oscillations** reported on the right panels in Fig. fig:rabi are in excellent agreement with theory....The transmission spectrum of the sample, measured with a vector network analyzer is reported in Fig. fig:tm6q(a). The seven absorption peaks correspond to the seven readout resonators. Close to each peak its bare resonance **frequency** as well the identification number of the device are printed. The inset of Fig. fig:tm6q(b) shows the transmission at **frequencies** around the resonance of device #3 vs. the magnetic flux bias. The two points at which the dispersive **frequency** shift changes its sign correspond to avoided level crossings of the **qubit** and resonator. We demonstrate FDM by measuring the maximum possible number of devices simultaneously. Figure fig:tm6q(b) shows the transmitted amplitude of the six probe tones versus the external uniformly applied magnetic flux. Each curve is shown aligned with the corresponding transmission peak to the left in Fig. fig:tm6q(a). The amplitude of the transmitted signal is constant as long as the **qubit** remains far detuned from the resonator. The amplitude changes drastically around two distinct fluxes, again indicating anti-crossings between the **qubit** and the corresponding resonator. There is a minimum between these two peaks, because the readout **frequencies** were set on resonance, with the dispersive shifts at the symmetry points of the **qubits** taken into account. The readout **frequency** of device #3 is shown as a dashed line in the inset....(color online). Simultaneous manipulation and detection of three **qubits**. Left plots: Rabi **oscillations** at several powers; traces are vertically offset for better visibility; curves with the same color/offset (blue-bottom, green-center, red-top) are measured simultaneously using the FDM technique described in the main text. Right plots: Rabi **oscillation** **frequency** versus power of the excitation tone; the error bars are smaller than the size of the dots....(color online). Multiplexed spectroscopy of **qubits** #2, 3 and 5. The **qubit** manipulation microwave excites **qubits** when its **frequency** matches the transition between their ground and excited states. The state of all three **qubits** is continuously and simultaneously monitored by the multi-tone probe signal. The horizontal axis reports the uniform bias flux applied to the chip....(color online). (a) Transmission spectrum of the sample with all **qubits** far detuned from the resonances. (b) FDM readout of six flux **qubits**. The main plot shows the transmission amplitude at the resonance **frequencies** of devices #1 to 6 vs. the magnetic flux generated by the uniform field coil, measured using FDM. The inset shows the transmission amplitude at several **frequencies** close to resonance #3, measured with a network analyzer. The dashed line indicates the probe **frequency** used for this device in the main plot. The curves in the main plot are normalized and shifted vertically for better visibility. The offset along the horizontal axis is due to magnetic field non-uniformity, which is likely due to the screening currents generated in the superconducting ground plane....superconducting flux **qubit**, **qubit** register, dispersive readout, **frequency** division
multiplexing, microwave resonators, Rabi **oscillations**
...Experimental setup used for FDM readout. The **qubit** manipulation signal is generated by a single microwave source for spectroscopy and three additional microwave sources, DAC channels and mixers for pulsed excitation of the **qubits**. ... An important desired ingredient of superconducting quantum circuits is a readout scheme whose complexity does not increase with the number of **qubits** involved in the measurement. Here, we present a readout scheme employing a single microwave line, which enables simultaneous readout of multiple **qubits**. Consequently, scaling up superconducting **qubit** circuits is no longer limited by the readout apparatus. Parallel readout of 6 flux **qubits** using a **frequency** division multiplexing technique is demonstrated, as well as simultaneous manipulation and time resolved measurement of 3 **qubits**. We discuss how this technique can be scaled up to read out hundreds of **qubits** on a chip.

Data types:

Contributors: Johansson, J., Saito, S., Meno, T., Nakano, H., Ueda, M., Semba, K., Takayanagi, H.

Date: 2005-10-17

In our sample geometry, the **qubit** is spatially separated from the rest of the circuitry. The **qubit** is enclosed by a superconducting quantum interference device (SQUID) that is inductively coupled to the **qubit**. The SQUID functions as a detector for the **qubit** state: the switching current of the SQUID is sensitive to the flux produced by the current in the **qubit**. The **qubit** is also enclosed by a larger loop containing on–chip capacitors that provide a well–defined electromagnetic environment for the SQUID and filtering of the measurement leads. The lead inductance L and capacitance C in the outer loop constitute an LC **oscillator** [see Fig. fig1(b)] with resonance **frequency** ω r = 2 π ν r = 1 / L C . The LC **oscillator** is described by a simple harmonic **oscillator** Hamiltonian: H o s c = ℏ ω r a † a + 1 / 2 , where a † ( a ) is the plasmon creation (annihilation) operator. The **qubit** is coupled to the LC **oscillator** via the mutual inductance M , giving an interaction Hamiltonian H I = h λ σ z a † + a , where the coupling constant is h λ = M I p ℏ ω r / 2 L . The total system is thus described by a Jaynes-Cummings type of Hamiltonian H = h / 2 ϵ σ z + Δ σ x + ℏ ω r a † a + 1 / 2 + h λ σ z a † + a . We denote the state of the system by | Q , i , with the **qubit** either in the ground ( Q = g ) or excited ( Q = e ) state, and the **oscillator** in the Fock state ( i = 0 , 1 , 2 , ). The parameters of the system can readily be engineered during fabrication; the **qubit** gap is determined by α and the junction resistance, the **oscillator** plasma **frequency** is fully determined by L and C , and the coupling between the **qubit** and the **oscillator** can be tuned by M and L ....fig1 (a) SEM micrograph of the sample. The **qubit** and the detector SQUID enclosing it are the small square loops in the lower center picture. The square plates at the top of the picture are the top plates of the on-chip capacitors separated by an insulator from the large bottom plate. (b) A close up of the **qubit** and the SQUID. The **qubit** dimension is 10.2 × 10.4 μ m 2 . (c) Equivalent circuit of the sample. The Josephson junctions are indicated by crosses: three in the inner **qubit** loop and two in the SQUID. The LC mode is indicated by the dashed line. The inductance and capacitances are calculated from the geometry to be L = 140 pH and C = 10 pF, and the **qubit** LC **oscillator** mutual inductance to be M = 5.7 pH. The current and voltage lines are filtered through a series combination of copper powder filters and lossy coaxial cables at mixing chamber temperature and on–chip resistors ( R V = 3 k Ω and R I = 1 k Ω )....fig4 (a) Rabi **oscillations** when a 2 ns long pulse with **frequency** ν e x = 4.35 GHz and amplitude A M W ∝ 10 P o s c / 20 is inserted between the π pulse and the shift pulse. (b) Measured Rabi **oscillations** at different drive powers (symbols), and a fit (solid curve) to ∑ n = 0 3 a n cos n + 1 Ω R t exp - Γ t with a 0 , , a 3 and Ω R as the only fitting parameters ( Γ is fixed from a fit to the -100 dBm curve). (c) The weights of the different **frequency** components a 0 , , a 3 obtained from the fit as a function of the drive amplitude (The red arrows show the position of the curves in (b). The error-bars indicate the errors obtained from the fitting procedure....fig2 (a) Spectroscopic characterization of the **qubit**–**oscillator** system showing the LC **oscillator** at ν r = 4.35 GHz and the **qubit** dispersion around the gap of Δ = 2.1 GHz. (b) A close up of the region around 4.35 GHz (indicated by the red square in (a) showing an avoided crossing. The lines are guides for the eye. (c) Schematic of the pulse sequence used to obtain the spectroscopy in (b): the **qubit** operating point is fixed at 10.5 GHz and via the MW line a shift pulse of variable height moves the operating point to the vicinity of 4.35 GHz. Two 50 ns MW pulses separated by 2 ns are added to the shift pulse. Here the second MW pulse is phase shifted by 180 ∘ . The phase shifted second pulse damps the **oscillations** in the LC circuit , and is crucial in terms of resolving the relatively weak **qubit** signal in this region. After the MW pulses the **qubit** state is measured by applying a measurement pulse to the SQUID (green curve). The spectroscopy in Fig. 2(a) was obtained with the same scheme, but without the second phase shifted pulse....Next we investigate the dynamics of the coupled system in the time domain. We performed a measurement cycle where we first excited the **qubit** and then brought the **qubit** and the **oscillator** into resonance where the exchange of a single energy quantum between the **qubit** and **oscillator** manifests itself as the vacuum Rabi **oscillation** | e , 0 ↔ | g , 1 (see Fig. 3). Figure. 3(c) is a schematic of the pulse sequence: We started by fixing the **qubit** operating point far from the resonance point [point 3 in Fig. fig3(c)] and prepared the **qubit** in the excited state by employing a π pulse. The π pulse was followed by a shift pulse, which brought the **qubit** into resonance with the **oscillator** for the duration of the shift pulse. After the shift pulse the **qubit** and the **oscillator** were brought back into off–resonance and the measurement pulse was applied to detect the state of the **qubit**. It is important to note that the rise time of the shift pulse, τ r i s e = 0.8 ns, is adiabatic with respect to both the **qubit** and the **oscillator**, τ r i s e > 1 / E 2 π / ω r , but non-adiabatic with respect to the coupling of the two systems, τ r i s e **oscillations** is thus different from that of normal Rabi **oscillations** where the system is driven by an external classical field and **oscillates** between two energy eigenstates. Also, in the normal Rabi **oscillations** the Rabi **frequency** is determined by the drive amplitude whereas the vacuum Rabi **oscillation** **frequency** is determined only by the system‘s intrinsic parameters. The observed Rabi **oscillations** are in excellent agreement with those calculated numerically [solid line in Fig. 3(a)]. The numerical calculation uses the total Hamiltonian and incorporates the effects of the decoherence of the **qubit** and the **oscillator**. The calculation was performed with the known **qubit** and LC **oscillator** parameters (obtained from spectroscopy and **qubit** experiments: **qubit** dephasing rate Γ φ = 0.1 GHz, **qubit** relaxation rate Γ e = 0.2 MHz, Δ = 2.1 GHz, ω r / 2 π = 4.35 GHz) and by treating the coupling constant and **oscillator** dephasing and relaxation rates as fitting parameters. From the fit we extracted the coupling constant λ = 0.2 GHz, **oscillator** dephasing rate Γ φ = 0.3 GHz, and relaxation rate Γ e = 0.02 GHz. The coupling constant extracted from the fit agrees well with that calculated with the mutual inductance λ = 0.216 GHz....We have observed the coherent exchange of a single energy quantum between a flux **qubit** and a superconducting LC circuit acting as a quantum harmonic **oscillator**. The exchange of an energy quantum is known as the vacuum Rabi **oscillations**: the **qubit** is **oscillating** between the excited state and the ground state and the **oscillator** between the vacuum state and the first excited state. We have also obtained evidence of level quantization of the LC circuit by observing the change in the **oscillation** **frequency** when the LC circuit was not initially in the vacuum state....fig3 (a) Vacuum Rabi **oscillations** (symbols) and a numerical fit (solid line). (b) The few lowest unperturbed and dressed energy levels when the system is in resonance. (c) The **qubit** energy level diagram and pulse sequence for the vacuum Rabi measurements. The π pulse (4.6 ns long at -16 dBm) on the **qubit** brings the system from state 1 to 2 and the shift pulse changes the flux in the **qubit** by Φ s h i f t , which, in turn, changes the operating point from 2 to 3 where the system undergoes free evolution between | e , 0 and | g , 1 at the vacuum Rabi **frequency** Ω R until the shift pulse ends and the system returns to the initial operating point where the state is measured to be either in 2 or 4. ... We have observed the coherent exchange of a single energy quantum between a flux **qubit** and a superconducting LC circuit acting as a quantum harmonic **oscillator**. The exchange of an energy quantum is known as the vacuum Rabi **oscillations**: the **qubit** is **oscillating** between the excited state and the ground state and the **oscillator** between the vacuum state and the first excited state. We have also obtained evidence of level quantization of the LC circuit by observing the change in the **oscillation** **frequency** when the LC circuit was not initially in the vacuum state.

Data types:

Contributors: Averin, Dmitri V., Rabenstein, Kristian, Semenov, Vasili K.

Date: 2005-10-27

(a) Equivalent circuit of the flux detector based on the Josephson transmission line (JTL) and (b) diagram of scattering of the fluxon injected into the JTL with momentum k by the potential U x that is controlled by the measured **qubit**. The fluxons are periodically injected into the JTL by the generator and their scattering characteristics (transmission and reflection coefficients t k , r k ) are registered by the receiver....Schematics of the QND fluxon measurement of a **qubit** which suppresses the effect of back-action dephasing on the **qubit** **oscillations**. The fluxon injection **frequency** f is matched to the **qubit** **oscillation** **frequency** Δ : f ≃ Δ / π , so that the individual acts of measurement are done when the **qubit** density matrix is nearly diagonal in the σ z basis, and the measurement back-action does not introduce dephasing in the **oscillation** dynamics. ... We suggest a new type of the magnetic flux detector which can be optimized with respect to the measurement back-action, e.g. for the situation of quantum measurements. The detector is based on manipulation of ballistic motion of individual fluxons in a Josephson transmission line (JTL), with the output information contained in either probabilities of fluxon transmission/reflection, or time delay associated with the fluxon propagation through the JTL. We calculate the detector characteristics of the JTL and derive equations for conditional evolution of the measured system both in the transmission/reflection and the time-delay regimes. Combination of the quantum-limited detection with control over individual fluxons should make the JTL detector suitable for implementation of non-trivial quantum measurement strategies, including conditional measurements and feedback control schemes.

Data types:

Contributors: Gustavsson, Simon, Bylander, Jonas, Yan, Fei, Forn-Díaz, Pol, Bolkhovsky, Vlad, Braje, Danielle, Fitch, George, Harrabi, Khalil, Lennon, Donna, Miloshi, Jovi

Date: 2012-01-30

(a) Spectrum of device B. The spectral line at 2 is the resonator, whereas the **qubit** tunnel coupling is Δ = 5.4 . (b) Rabi **frequency** vs bias current , measured at f = 5.4 and = 0 and for two different microwave drive amplitudes . Similar to device A, the Rabi **frequency** depends strongly on , and scales linearly with drive amplitude. The black lines are fits to Eqs. ( eq:drive, eq:Rabi), using the same coupling parameters for both sets of data. Note that the range of in fig:RabiLongT1(b) is several times larger than in fig:Rabi(a)....We have investigated two devices with similar layouts but slightly different parameters, both made of aluminum. Device A was designed and fabricated at MIT Lincoln Laboratory and device B was designed and fabricated at NEC. Figure fig:Sample(b) shows a spectroscopy measurement of device A versus applied flux, with the **qubit** flux detuning defined as = Φ + Φ 0 / 2 and Φ 0 = h / 2 e . The **qubit** **frequency** follows = Δ 2 + ε 2 , where the tunnel coupling Δ = 2.6 is fixed by fabrication and the energy detuning ε = 2 I P / h is controlled by the applied flux Φ ( I P is the persistent current in the **qubit** loop). The resonator **frequency** is around 2.3 and depends only weakly on and . In addition, there are features visible at **frequencies** corresponding to the sum of the **qubit** and resonator **frequencies**, illustrating the coherent coupling between the two systems ....(a) Rabi **frequency** of **qubit** A, measured vs at = 0 . The driving field seen by the **qubit** contains two components: one is due to direct coupling to the antenna, the other is due to the coupling mediated by the resonator. (b) Rabi traces for a few of the data points in panel (a). The microwaves in the antenna have the same amplitude and **frequency** for all traces. (c) Direct coupling between the antenna and the **qubit**, extracted from measurements similar to the one shown in panel (a). The coupling depends only weakly on **frequency**. (d) Microwave current in the resonator, induced by a fixed microwave amplitude in the antenna. The black line is a fit to the square root of a Lorentzian, describing the **oscillation** amplitude of a harmonic **oscillator** with f r = 2.3 and Q = 100 ....Figures fig:Rabi(c) and fig:Rabi(d) show how the two drive components depend on microwave **frequency**, measured by changing the static flux detuning to increase the **qubit** **frequency** [see...(a) Circuit diagram of the **qubit** and the **oscillator**. The **qubit** state is encoded in currents circulating clockwise or counterclockwise in the **qubit** loop (blue arrow), while the mode of the harmonic **oscillator** is shown by the red arrows. (b) Spectrum for device A, showing the **qubit** and the harmonic **oscillator**. In addition, the two-photon **qubit** ( / 2 ) and the **qubit**+resonator ( + ) transitions are visible. (c) Flux induced in the **qubit** loop by the dc bias current . The black lines are parabolic fits. (d) First-order coupling between the **qubit** and the ground state of the harmonic **oscillator**, showing that the coupling is tunable by adjusting . The coupling is zero at = * , which is slightly offset from = 0 due to fabricated junction asymmetry. The derivative ε is calculated from the curves in panel (c). The **qubit** parameters are: I P = 175 for device A and I P = 180 for device B. The resonators have quality factors Q ≈ 100 . The right-hand axis is calculated using = 2.2 and C e f f = 2 C = 14 for both samples....Having determined the coupling coefficients, we turn to analyzing how the presence of the resonator influences the ** qubit’s** driven dynamics. Figure fig:Rabi(a) shows the extracted Rabi

**frequency**of

**qubit**A as a function of , measured at = 2.6 . We find that changes by a factor of five over the range of the measurement, which is surprising since both the amplitude and the

**frequency**of the microwave current in the antenna are kept constant. The data points were obtained by fitting Rabi

**oscillations**to decaying sinusoids, a few examples of Rabi traces for different values of are shown in...(a) Decay envelopes of the Rabi and rotary-echo sequences for device B, measured with = 65 . The solid lines are fits to eq:f. (b) Decay times for Rabi and rotary echo, extracted from fits similar to the ones shown in panel (a). The dashed line shows the upper limit set by

**qubit**energy relaxation. The dotted line marks the position for the decay envelope shown in panel (a). (c,d) Schematic diagrams describing the two pulse sequences in (a) and (b). For rotary echo, the phase of the microwaves is rotated by 180 ∘ during the second half of the sequence....We have investigated the driven dynamics of a superconducting flux

**qubit**that is tunably coupled to a microwave resonator. We find that the

**qubit**experiences an

**oscillating**field mediated by off-resonant driving of the resonator, leading to strong modifications of the

**qubit**Rabi

**frequency**. This opens an additional noise channel, and we find that low-

**frequency**noise in the coupling parameter causes a reduction of the coherence time during driven evolution. The noise can be mitigated with the rotary-echo pulse sequence, which, for driven systems, is analogous to the Hahn-echo sequence....To further investigate how the presence of the resonator affects the

**qubit**dynamics at large detunings, we performed measurements on device B. Figure fig:RabiLongT1(a) shows a spectrum of that device, where the

**qubit**and the resonator mode ( f r = 2 ) are clearly visible. This device has a larger tunnel coupling ( Δ = 5.4 ), which allows us to operate the

**qubit**at large

**frequency**detuning from the resonator while still staying at ε d c = 0 , where the

**qubit**, to first order, is insensitive to flux noise . The

**qubit**-resonator detuning corresponds to several hundred linewidths of the resonator, which is the regime of most interest for quantum information processing ....Figure fig:RabiLongT1(b) shows the Rabi

**frequency**vs bias current of device B, measured at f = 5.4 and for two different values of the microwave drive current . Similarly to ... We have investigated the driven dynamics of a superconducting flux

**qubit**that is tunably coupled to a microwave resonator. We find that the

**qubit**experiences an

**oscillating**field mediated by off-resonant driving of the resonator, leading to strong modifications of the

**qubit**Rabi

**frequency**. This opens an additional noise channel, and we find that low-

**frequency**noise in the coupling parameter causes a reduction of the coherence time during driven evolution. The noise can be mitigated with the rotary-echo pulse sequence, which, for driven systems, is analogous to the Hahn-echo sequence.

Data types:

Contributors: Leyton, V., Thorwart, M., Peano, V.

Date: 2011-09-26

(Color online) (a) Nonlinear response A of the detector coupled to the **qubit** prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for the same parameters as in Fig. fig2. The quadratic **qubit**-detector coupling induces a global **frequency** shift of the response by δ ω e x = 2 g . (b) Discrimination power D ω e x of the detector coupled to the **qubit** for the same parameters as in a). fig3...(Color online) (a) Asymptotic population difference P ∞ of the **qubit** states, and (b) the corresponding detector response A as a function of the external **frequency** ω e x for the same parameters as in Fig. fig2. fig4...(Color online) (a) Relaxation rate Γ of the nonlinear quantum detector, (b) the measurement time T m e a s , and (c) the measurement efficiency Γ m e a s / Γ as a function of the external **frequency** ω e x . The parameters are the same as in Fig. fig2. fig5...For a fixed value of g , the shift between the two cases of the opposite **qubit** states is given by the **frequency** gap δ ω e x ≃ 2 g . Figure fig3 (a) shows the nonlinear response of the detector for the two cases when the **qubit** is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line)....(Color online) (a) Amplitude A of the nonlinear response of the decoupled quantum Duffing detector ( g = 0 ) as a function of the external driving **frequency** ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote the corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1...We introduce a detection scheme for the state of a **qubit**, which is based on resonant few-photon transitions in a driven nonlinear resonator. The latter is parametrically coupled to the **qubit** and is used as its detector. Close to the fundamental resonator **frequency**, the nonlinear resonator shows sharp resonant few-photon transitions. Depending on the **qubit** state, these few-photon resonances are shifted to different driving **frequencies**. We show that this detection scheme offers the advantage of small back action, a large discrimination power with an enhanced read-out fidelity, and a sufficiently large measurement efficiency. A realization of this scheme in the form of a persistent current **qubit** inductively coupled to a driven SQUID detector in its nonlinear regime is discussed....Before turning to the quantum detection scheme, we discuss the dynamical properties of the isolated detector, which is the quantum Duffing **oscillator**. A key property is its nonlinearity which generates multiphoton transitions at **frequencies** ω e x close to the fundamental **frequency** Ω . In order to see this, one can consider first the undriven nonlinear **oscillator** with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , the degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in the detector. As a consequence, the amplitude A of the nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small **oscillation** state is predominantly populated. The formation of peaks and dips goes along with jumps in the phase of the **oscillation**, leading to **oscillations** in or out of phase with the driving. A typical example of the nonlinear response of the quantum Duffing **oscillator** in the deep quantum regime containing few-photon (anti-)resonances is shown in Fig. fig1(a) (decoupled from the **qubit**), together with the corresponding quasienergy spectrum [Fig. fig1(b)]. We show the multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined by the Rabi **frequency**, which is given by the minimal splitting at the corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to the driving strength f , one can get the minimal energy splitting at the avoided quasienergy level crossing 0 N as...(Color online) Nonlinear response A of the detector as a function of the external driving **frequency** ω e x in the presence of a finite coupling g = 0.0012 Ω to the **qubit** (black solid line). The blue dashed line indicates the response of the isolated detector. The parameters are the same as in Fig. fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2...Notice that g and α depend on the external flux ϕ e x , i.e., they are tunable in a limited regime with respect to the desired **oscillator** **frequency** Ω , where the coupling term is considered as a perturbation to the SQUID ( g **oscillator** to dominate. The dependence of the dimensionless ratios α / Ω and g / Ω is shown in Fig. fig0. ... We introduce a detection scheme for the state of a **qubit**, which is based on resonant few-photon transitions in a driven nonlinear resonator. The latter is parametrically coupled to the **qubit** and is used as its detector. Close to the fundamental resonator **frequency**, the nonlinear resonator shows sharp resonant few-photon transitions. Depending on the **qubit** state, these few-photon resonances are shifted to different driving **frequencies**. We show that this detection scheme offers the advantage of small back action, a large discrimination power with an enhanced read-out fidelity, and a sufficiently large measurement efficiency. A realization of this scheme in the form of a persistent current **qubit** inductively coupled to a driven SQUID detector in its nonlinear regime is discussed.

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